Determine how many solutions exist for the system of equations. ${-2x+2y = -4}$ ${y = -2+x}$
Answer: Convert both equations to slope-intercept form: ${-2x+2y = -4}$ $-2x{+2x} + 2y = -4{+2x}$ $2y = -4+2x$ $y = -2+x$ ${y = x-2}$ ${y = -2+x}$ ${y = x-2}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = x-2}$ ${y = x-2}$ Both equations have the same slope and the same y-intercept, which means the lines would completely overlap. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Since any solution of ${-2x+2y = -4}$ is also a solution of ${y = -2+x}$, there are infinitely many solutions.